:PROPERTIES: :ID: 0a1fdfe8-2e52-431b-808b-412660a8e9b0 :END: #+TITLE: Linear Algebra Done Right #+LATEX_COMPILER: lualatex #+LATEX_CLASS: article #+LATEX_CLASS_OPTIONS: [a4paper, 11pt] #+OPTIONS: title:nil toc:nil num:t author:nil date:nil ^:{} #+LATEX_HEADER: \setlength{\headheight}{13.59999pt} #+LATEX_HEADER: \definecolor{linkblue}{HTML}{00007B} #+LATEX_HEADER: \definecolor{exampleblue}{HTML}{4169E1} #+LATEX_HEADER: \definecolor{highlightred}{HTML}{B22222} #+LATEX_HEADER: \definecolor{highlightblue}{HTML}{0000FF} #+LATEX_HEADER: \definecolor{notegray}{HTML}{7F807F} #+LATEX_HEADER: \definecolor{highlightsalmon}{HTML}{F08080} #+LATEX_HEADER: \definecolor{highlightgreen}{HTML}{006400} #+LATEX_HEADER: \definecolor{examplebarbg}{rgb}{0.255, 0.41, 0.884} #+LATEX_HEADER: \usepackage{amsthm} #+LATEX_HEADER: \usepackage{mathtools} #+LATEX_HEADER: \usepackage{thmtools} #+LATEX_HEADER: \usepackage{mathrsfs} #+LATEX_HEADER: \usepackage{bm} #+LATEX_HEADER: \usepackage{fancyhdr} #+LATEX_HEADER: \pagestyle{fancy} #+LATEX_HEADER: \fancyhf{} #+LATEX_HEADER: \lhead{\textit{Aayush Bajaj}} #+LATEX_HEADER: \rhead{\thepage} #+LATEX_HEADER: \renewcommand{\headrulewidth}{0pt} #+LATEX_HEADER: \usepackage{tikz} #+LATEX_HEADER: \usepackage{caption} #+LATEX_HEADER: \usetikzlibrary{arrows.meta, positioning, shapes.geometric, calc} #+LATEX_HEADER: \usepackage{mdframed} #+LATEX_HEADER: \usepackage[bottom]{footmisc} #+LATEX_HEADER: \setlength{\parskip}{0pt plus 1pt} #+LATEX_HEADER: \usepackage{titlesec} #+LATEX_HEADER: \titleformat{\section}{\normalfont\normalsize\centering}{\thesection}{1em}{\scshape} #+LATEX_HEADER: \titleformat{\subsection}{\normalfont\bfseries\fontsize{12}{14.4}\selectfont}{\thesubsection}{1em}{} #+LATEX_HEADER: \titleformat{\subsubsection}{\normalfont\bfseries}{\thesubsubsection}{1em}{} #+LATEX_HEADER: \newtheoremstyle{italicbody}{3pt}{3pt}{\itshape}{}{\bfseries}{.}{ }{} #+LATEX_HEADER: \newtheoremstyle{uprightbody}{3pt}{3pt}{}{}{\bfseries}{.}{ }{} #+LATEX_HEADER: \theoremstyle{italicbody} #+LATEX_HEADER: \newtheorem{theorem}{Theorem}[subsection] #+LATEX_HEADER: \newtheorem{lemma}{Lemma}[subsection] #+LATEX_HEADER: \newtheorem{corollary}{Corollary}[subsection] #+LATEX_HEADER: \newtheorem{proposition}{Proposition}[subsection] #+LATEX_HEADER: \theoremstyle{uprightbody} #+LATEX_HEADER: \newtheorem{definition}{Definition}[subsection] #+LATEX_HEADER: \newtheorem{examplesinner}{Examples}[subsection] #+LATEX_HEADER: \newtheorem{remark}{Remark}[subsection] #+LATEX_HEADER: \newtheorem{result}{Result}[subsection] #+LATEX_HEADER: \newmdenv[leftline=true,rightline=false,topline=false,bottomline=false,linewidth=3pt,linecolor=examplebarbg,innerleftmargin=10pt,innerrightmargin=0pt,innertopmargin=2pt,innerbottommargin=2pt,skipabove=3pt,skipbelow=3pt]{examplebox} #+LATEX_HEADER: \newenvironment{examples}[1][]{\begin{examplebox}\refstepcounter{examplesinner}{\color{exampleblue}\sffamily\bfseries Examples~\theexamplesinner\ifx\\#1\\\else\ (#1)\fi.}\,}{\end{examplebox}} #+LATEX_HEADER: \renewcommand{\theHtheorem}{\thesection.\thesubsection.\arabic{theorem}} #+LATEX_HEADER: \renewcommand{\theHlemma}{\thesection.\thesubsection.\arabic{lemma}} #+LATEX_HEADER: \renewcommand{\theHcorollary}{\thesection.\thesubsection.\arabic{corollary}} #+LATEX_HEADER: \renewcommand{\theHproposition}{\thesection.\thesubsection.\arabic{proposition}} #+LATEX_HEADER: \renewcommand{\theHdefinition}{\thesection.\thesubsection.\arabic{definition}} #+LATEX_HEADER: \renewcommand{\theHexamplesinner}{\thesection.\thesubsection.\arabic{examplesinner}} #+LATEX_HEADER: \renewcommand{\theHremark}{\thesection.\thesubsection.\arabic{remark}} #+LATEX_HEADER: \renewcommand{\theHresult}{\thesection.\thesubsection.\arabic{result}} #+LATEX_HEADER: \newcommand{\E}{\mathbb{E}} #+LATEX_HEADER: \newcommand{\Var}{\operatorname{Var}} #+LATEX_HEADER: \newcommand{\Cov}{\operatorname{Cov}} #+LATEX_HEADER: \renewcommand{\P}{\mathbb{P}} #+LATEX_HEADER: \newcommand{\Q}{\mathbb{Q}} #+LATEX_HEADER: \newcommand{\R}{\mathbb{R}} #+LATEX_HEADER: \newcommand{\N}{\mathcal{N}} #+LATEX_HEADER: \newcommand{\F}{\mathcal{F}} #+LATEX_HEADER: \newcommand{\G}{\mathcal{G}} #+LATEX_HEADER: \newcommand{\A}{\mathcal{A}} #+LATEX_HEADER: \newcommand{\B}{\mathcal{B}} #+LATEX_HEADER: \newcommand{\Sc}{\mathcal{S}^c} #+LATEX_HEADER: \newcommand{\Lp}{\mathcal{L}} #+LATEX_HEADER: \newcommand{\indic}{\mathbf{1}} #+LATEX_HEADER: \DeclareMathOperator{\sgn}{sgn} #+LATEX_HEADER: \newcommand{\hblue}[1]{{\color{highlightblue}#1}} #+LATEX_HEADER: \newcommand{\hred}[1]{{\color{highlightred}#1}} #+LATEX_HEADER: \newcommand{\hgray}[1]{{\color{notegray}#1}} #+LATEX_HEADER: \newcommand{\hsalmon}[1]{{\color{highlightsalmon}#1}} #+LATEX_HEADER: \newcommand{\hgreen}[1]{{\color{highlightgreen}#1}} #+LATEX_HEADER: \renewcommand{\qedsymbol}{$\blacksquare$} #+LATEX_HEADER: \pagestyle{empty} #+BEGIN_EXPORT latex \geometry{a4paper, top=2cm, left=1cm, right=1cm} \pagestyle{fancy} \hypersetup{colorlinks=true, linkcolor=linkblue, citecolor=linkblue, urlcolor=linkblue} \begin{center} {\bfseries \large Linear Algebra Done Right} \\[0.5em] February 2026 \end{center} \tableofcontents \newpage #+END_EXPORT * Vector Spaces * Finite-Dimensional Vector Spaces * Linear Maps * Polynomials * Eigenvalues and Eigenvectors * Inner Product Spaces * Operators on Inner Product Spaces * Operators on Complex Vector Spaces * Multilinear Algebra and Determinants $U=\{(x,0,0)\in\mathrm{{F}}^{3}:x\in\mathrm{{F}}\}$